The -Cohomology Rings of -Manifolds

Abstract

We compute the rings H * (N; F 2 ) for N a closed Sol 3 -manifold, and then determine the Borsuk-Ulam indices BU(N, φ) with φ 0 in H 1 (N; F 2 ).2010 Mathematics subject classification: primary 57M99.

-Manifolds

“…If A has order 2 then det(A) = −1, while if A has order 4 then det(A) = +1 and A 2 = −I, and similarly for B. Thus if F = 1 then (3) holds, while if F = {±I} then (4), (5)…”

“…It is easy to see that every flat 3-manifold group or Sol 3 -group can be generated by at most three elements [5], and hence that every Sol 3 × E 1 -group requires at most four generators. This is best possible in general.…”

The cohomology ring of the sapphires that admit the Sol geometry

“…Now we determine H * (G; Z p ) when p is an odd prime. The case p = 2 has been solved by Hillman [Hil14], but could also be recovered from our techniques.…”

Diagonal approximation and the cohomology ring of torus fiber bundles

“…We observe also that J. Hillman has recently determined the cohomology ring of the Sol 3 -manifolds with coefficients Z2 using different methods. See [3].…”